Depending on interpretation of QM, can Hilbert space be...

Depending on interpretation of QM, can hilbert space be considered just as real as space time? In MWI the wave function is real, but still lies in hilbert space, so would hilbert space be considered a real space according to this interpretation?

Staff: Mentor

A Hilbertspace H is a real or complex vectorspace with a scalar product on its elements. The latter induces a norm and therefore a metric. H is also complete related to this metric, i.e. "all limits exists" (not quite precise, but essentially what it means).
So you can imagine H as an ordinary euclidean vectorspace over the reals if you want.

A Hilbertspace H is a real or complex vectorspace with a scalar product on its elements. The latter induces a norm and therefore a metric. H is also complete related to this metric, i.e. "all limits exists" (not quite precise, but essentially what it means).
So you can imagine H as an ordinary euclidean vectorspace over the reals if you want.

So is it considered a real (physical) space like space time in theories like MWI (where the wave function is considered real)? I think i read on a few articles that hilbert space is actually considered the fundamental physical space in some of the Ψ ontic interpretations.

Staff: Mentor

So is it considered a real (physical) space like space time in theories like MWI (where the wave function is considered real)? I think i read on a few articles that hilbert space is actually considered the fundamental physical space in some of the Ψ ontic interpretations.

I think it doesn't matter whether you have ℝ or ℂ as underlying field. For computing purposes, e.g. the Schrödinger equations, it's usually easier to use complex numbers. A Hilbert space is nothing special. In this context it's just a natural environment since it supplies angels, distances, operators, lengthes.

I think it doesn't matter whether you have ℝ or ℂ as underlying field. For computing purposes, e.g. the Schrödinger equations, it's usually easier to use complex numbers. A Hilbert space is nothing special. In this context it's just a natural environment since it supplies angels, distances, operators, lengthes.

Staff: Mentor

I have a feeling that you two are talking past each other. What the OP means by being "real" is that they are, to use the distinction the OP alludes to, ontological i.e. really out there as opposed to epistemic like probabilities or beliefs that are only in your head and do not have to match what happens out there. In the latter view, Hilbert space would just be a bookkeeping, calculational tool. This has nothing to do with real numbers, which I suppose is what fresh_42 thinks about. (Well, are real numbers real?)

To the OP, since a vector space is just a collection of vectors (in this case the wave functions), if the vectors are taken to be real then the vector space should also be real. So I suppose the Hilbert space is real in those interpretations too. Although I don't know how to think of, say, an inner product as real.

Staff: Mentor

Depending on interpretation of QM, can hilbert space be considered just as real as space time?

In some interpretations elements of the QM Hilbert space is considered real. There is a technical complication though in that the space talked about in physics texts is not really a Hilbert space - but what's called a Rigged Hilbert Space (RHS). The test spaces of the RHS are what would be considered real. But since these are chosen depending on the problem it makes such a view somewhat more difficult - although you can probably formulate one (I can think of at least one way) - I think it suggests it probably isn't real.